Rinse the strawberries, remove the stems and any soft or brown spots, and cut into quarters. Working in batches, crush them well with a masher or sturdy spoon.
Combine the crushed strawberries in a large pot with the water. Bring to a boil over medium-high heat, then reduce heat, cover, and simmer 10 minutes. Remove from heat, skim off any foam, and let sit uncovered for 15 minutes.
Set a fine-mesh sieve over a large bowl and strain the cooked berries in batches. For a clearer syrup, strain again through 3 to 4 layers of dampened cheesecloth, then once more through a paper coffee filter. Discard the pulp.
Measure the strawberry juice into a clean saucepan. If short, top up with a splash of water or mild juice like apple or white grape.
Add the optional lemon juice and the sugar. Heat over medium-high, stirring, until the sugar dissolves. Bring to a boil, then reduce heat and simmer 5 minutes. Remove from heat and skim off any foam.
Ladle the hot syrup into prepared jars, leaving 1/4 inch headspace. Wipe the rims, apply lids and bands, and tighten to fingertip tight.
Lower the filled jars into a prepared water bath canner, covered by at least 1 inch of water. Process for 10 minutes for pints or half pints, adjusting for altitude. If not canning, cool and then refrigerate or freeze.
Turn off the heat and let the jars stand in the canner for 5 minutes before removing them. Cool undisturbed for 12 to 24 hours, check seals, and store. Refrigerate any jars that didn't seal for immediate use.